3.2 Example: coin toss

The outcome of a coin toss is uncertain in most cases. We can consider some of the questions from the previous section to describe this uncertainty.

What uncertainties are there?

The uncertainty is whether the coin will land heads-up or tails-up as a result of the toss.

What are the sources of this uncertainty?

• the coin toss process
  • aleatory uncertainty: there is randomness to the coin toss process
  • epistemic uncertainty: is the person tossing the coin able to do so without favouring one side of the coin over the other? This can be reduced by learning about the person tossing the coin, e.g. from previous tosses.
• potential double sided coin
  • epistemic uncertainty: can be eliminated by checking both sides of the coin

Assume that the coin is checked and has one head and one tail, and that the person tossing the coin is doing so fairly. There is now only aleatory uncertainty about the outcome. This is an irreducible uncertainty about the coin toss.

What is the magnitude of uncertainty?

The coin toss has only two possible outcomes, head or tail. The probability that one of these outcomes must occur is 1, since it is certain. This means that the events are exhaustive, i.e. the events exhaust all possible outcomes.

A head and a tail cannot occur together and so they are also known as mutually exclusive. When outcomes are mutually exclusive, then their probabilities can be added together - this is one of the laws of probability. Because both outcomes of the coin toss are also exhaustive then this means that whatever the probability of the individual outcomes, they must sum to 1.

If a head and a tail are equally likely, then due to the rules above they must each have a probability equal to 0.5. This quantified uncertainty is the result of the logic of probability. These values form a probability model for the outcomes of the coin toss. This probability model can be checked to confirm that it aligns with specified beliefs. For example, since we hold the belief that each probability is 0.5, then we expect that the outcomes of multiple coin tosses should be evenly distributed between heads and tails.

Notice that if we did not ignore some of the epistemic uncertainties about the coin toss (e.g. if we did not trust the coin flipper to be fair), then the probability model would need to be adjusted to align with this belief, e.g. by changing the probability of heads from 0.5.